Directions: In the following question, two statements are numbered as

Directions: In the following question, two statements are numbered as Quantity I and Quantity II. On solving these statements, we get quantities I and II respectively. Solve both quantities and choose the correct option.Quantity I: Find the total surface area of a hemisphere where the total surface area of the sphere is 3 times the curved surface area of a cone whose radius and height are 6 cm and 8 cm respectively. (take π = 3.14)Quantity II: Find the total surface area of the cuboid whose length, height, and breadth are in the ratio 5 ∶ 2 ∶ 3. Given the difference between the length and bredth is 5 cm. 1. Quantity I > Quantity II2. Quantity I < Quantity II3. Quantity I ≤ Quantity II4. Quantity I ≥ Quantity II5. Quantity I = Quantity II or relation cannot be established.

Answer» Correct Answer - Option 1 : Quantity I > Quantity II

Given:

Quantity I:

Radius and height of the cone = 6 cm and 8 cm

The total surface area of sphere = 3 × curved surface area of the cone

Quantity II:

The ratio of length, height, and breadth of cuboid = 5 ∶ 2 ∶ 3

Difference between length and height = 5 cm

Concept used:

The volume of sphere = 4/3 × π × r³

The total surface area of sphere = 4πr²

The curved surface area of cone = π × r × l

Where l = √(r² + h²)

Total surface area of the cuboid = 2 × (lb + bh + hl)

Where, l, b, and h are the length, breadth, and height.

Calculation:

Quantity I:

l = √(6² + 8²) = 10 cm

curved surface area of cone = π × 6 × 10 = 60π

total surface area of sphere = 4πr² = 3 × 60π

⇒ r² = 45 cm²

Total surface area of the hemisphere = 3πr² = 3π × 45

Total surface area of the hemisphere = 135 × 3.14

Quantity I = 135 × 3.14 = 423.9 cm²

Quantity II:

Ratio of length, height and breadth of cuboid = 5 ∶ 2 ∶ 3

Then, length, breadth and height of cuboid will be = 5x, 3x and 2x

Difference between length and breadth = 5x – 3x = 2x

⇒ 2x = 5

Or, x = 5/2 cm

Total surface area of the cuboid = 2 × (15x² + 6x² + 10x²) = 62 × (5/2)²

Quantity II = 62 × (5/2)² = 387.5 cm²

Quantity I	Relation	Quantity II
423.9 cm²	>	387.5 cm²

∴ From the table, we can see that Quantity I > Quantity II.